# Urysohn's Metrization Theorem

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## Theorem

Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.

Then $T$ is metrizable.

## Proof

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## Also see

- Metrizable Space is not necessarily Second-Countable, indicating that the converse does not hold.

## Source of Name

This entry was named for Pavel Samuilovich Urysohn.

## Historical Note

This form of **Urysohn's Metrization Theorem** was actually proved by Andrey Nikolayevich Tychonoff in 1926.

What Urysohn had shown, in a posthumous $1925$ paper, was that every second-countable *normal* Hausdorff space is metrizable.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metrizability